Automorphic Numbers with 10 Digits

Theorem

The only $10$-digit automorphic numbers are:

$1 \, 787 \, 109 \, 376$

$8 \, 212 \, 890 \, 625$

Proof

We have:

\(\ds 1 \, 787 \, 109 \, 376^2\)

\(=\)

\(\ds \enspace 3 \, 193 \, 759 \, 92 \mathbf {1 \, 787 \, 109 \, 376}\)

\(\ds 8 \, 212 \, 890 \, 625^2\)

\(=\)

\(\ds 67 \, 451 \, 572 \, 41 \mathbf {8 \, 212 \, 890 \, 625}\)

thus demonstrating they are automorphic.

By Automorphic Numbers in Base 10, there are no others.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1,787,109,376$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1,787,109,376$